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[U]: Undergraduate

[G]: Graduate

[U] Andrews: Number Theory — A good introductory text with some unusual combinatorial proofs.

[U] Apostol; Introduction to Analytic Number Theory — A good introduction to both analytic number theory and to many topics which the PROMYS number theory covers lightly. Some familiarity with calculus is assumed and the last few chapters assume some knowledge of complex analysis.

[U/G] Borevich & Shafarevich: Number Theory — A beautifully written “introduction” to the more advanced aspects of modern algebraic number theory. Hard to read without a strong background in such things.

[G] Cohen: A Course in Computational Algebraic Number Theory — Your number one source for algorithms in computational algebraic number theory.

[U] Cohn: Advanced Number Theory — This book, as its name suggests, is fairly advanced but it is quite accessible. It is a good book to read after PROMYS, especially if Z[√-5] upsets you.

[U] Conway, John H.: The Sensual (Quadratic) Form — A unique approach to quadratic forms. Includes unusual proofs of Hasse-Minkowski, quadratic reciprocity, and the three squares theorem.

[U/G] Cox, David: Primes of the form x^2+ny^2 -- A wonderful book that introduces such ideas as complex multiplication, class field theory, elliptic curves, and binary quadratic forms, in a down to earth and very accessible way! Lots of fun.

[U] Davenport: The Higher Arithmetic, 7th Ed., Cambridge — A good book to read after PROMYS.

[G] Davenport: Multiplicative Number Theory – Good introduction to Dirichlet’s theorem, the zeta function, and sieve theory.

[U] Edwards, H. M.: Riemann’s Zeta Function — A great description of the zeta function and what is known about it; assumes complex analysis. Now reprinted by Dover, so you have no excuse for not owning it.

[U] Gouvea. p-adic numbers — Very accessible. Also serves as an excellent tutorial on how to read harder math books in general.

[U] Hardy and Wright: An Introduction to the Theory of Numbers — A thorough and classical treatment. Contains a surprising wealth of information.

[U] Ireland and Rosen: A Classical Introduction to Modern Number Theory — Covers the rudiments of algebraic number theory concretely using mostly elementary methods. Includes a treatment of cubic and biquadratic reciprocity.

[U] Gelfand: The Solution of Equations in Integers —A good review of some of the topics covered this summer plus a glimpse of what lies beyond.

[G] Janusz: Algebraic Number Fields — A readable introduction to class field theory for someone who knows Galois theory; technically it requires no knowledge of algebraic number theory, but a previous book on the topic may help.

[U] Khinchin: Continued Fractions – An extensive treatment, elementary in the beginning, but difficult toward the end.

[U] Khinchin: Three Pearls of Number Theory — A very beautiful little book, but difficult.

[G] Koblitz: p-adic Numbers, p-adic Analysis, and Zeta-Functions — A very good and readable introduction to p-adics. Feel free to skip chapter two, as it is far more advanced than the rest of the book and is not used later.

[U/G] Lekkerkerker: Geometry of Numbers — With a name like Lekkerkerker, how can you go wrong? The standard, comprehensive treatment of Minkowski theory. Generalizes and applies Minkowski’s theorem in enough ways to kill a horse.

[U] Marcus: Number Fields. — One of the most readable introductions to algebraic number theory. It’s good to have seen rings and some Galois theory before.

[G] Neukirch: Algebraic Number Theory – An extremely comprehensive, streamlined and sophisticated resource, including a treatment of axiomatic class field theory and the Grothendieck-Riemann-Roch theorem; requires a lot of background to appreciate. Also includes a sizeable section on zeta functions and analytic number theory (but does not include the prime number theorem).

[G] Patterson: An Introduction to the Theory of the Riemann Zeta Function – A modern approach to the Zeta function with a view towards generalizations.

[U] Ribenboim: The New Book of Prime Number Records 3rd ed. – A great survey of everything you could want to know about prime numbers in Z, both current knowledge and history.

[U/G] Rosen: Number Theory in Function Fields — Everything you might ever want to know about Zp[x] and its relatives, including generalizations of Quadratic Reciprocity, Zeta Functions and more. There are few formal prerequisites, but you should be used to reading math books.

[U/G] Serre: A Course in Arithmetic — A pretty and quick introduction to p-adics, Dirichlet series, and modular forms – after you are comfortable with abstract algebra, topology, and complex analysis.

[U] Shanks: Solved and Unsolved Revised Problems in Number Theory, 4th Ed., Chelsea Publishers. — An unusual introduction to number theory including many relatively recent results and conjectures.

[U] Silverman & Tate: Rational Points on Elliptic Curves — A very computational elementary introduction to the arithmetic on elliptic curves.

[G] Silverman: The Arithmetic of Elliptic Curves – A standard reference for the basic theory of elliptic curves.

[G] Silverman: Advanced Topics in the Arithmetic of Elliptic Curves – A sequel to the above; includes more specialized topics such as complex multiplication and Neron models.

[U] Stopple: A Primer of Analytic Number Theory — This is a comfortable introduction to analytic number theory which does not assume much more than basic calculus. Topics include the zeta function, Pell’s equation, binary quadratic forms and the class formula.

[G] Swinnerton-Dyer: A Brief Guide to Algebraic Number Theory — A gem of a book, compact and streamlined. A slick introduction to modern number theory.

[U/G] Thomas: Zeta Functions: An Introduction to Algebraic Geometry — Introduces algebraic geometry and the Weil conjectures using the zeta function of varieties over finite fields. Frighteningly elementary.

[U] Uspensky & Heaslet. Elementary Number Theory — Contains many good examples.

[U] Herstein: Topics in Algebra — A good place to start; excellent on groups, rings, fields, and linear algebra with very good problems. His newer book, Abstract Algebra, is more elementary but also very readable.

[U] Halmos: Finite Dimensional Vector Spaces — Excellent abstract treatment of linear algebra. Better than Hoffman and Kunze. Use this book.

[U/G] Atiyah & Macdonald: Commutative Algebra. — A short introduction to the algebra needed in algebraic geometry. College-level. Many good problems.

[U] Artin, Emil: Galois Theory — A beautiful presentation. Ridiculously slick; there are no problems, so take care to redo all the proofs.

[U] Artin, Michael: Algebra — A classic. Has a wonderful collection of problems. Highly recommended.

[G] Bourbaki: Algebra, Commutative Algebra, Integration, Topological Vector Spaces, etc. — These are technically parts of the same volume, but they are so massive they each come in their own book. Bourbaki does everything in the utmost generality. It can be unwieldy, but the algebra books are done quite nicely, and a lot of the material is not found anywhere else because no one else wants to write foundational books.

[U] Burn: Groups, a Path to Geometry — Lots of applications.

[U] Cox, Little & O’Shea: Ideals, Varieties and Algorithms - A very accessible introduction to basic ideas from computational algebraic geometry.

[G] Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry — A very comprehensive commutative algebra book with loads of excellent exercises. Too unwieldy to be used as an introduction, but an excellent place to go for all sorts of interesting, modern commutative algebra.

[U] Edwards: Galois Theory — Includes a translation of Galois’ writing and a decent history of the topic.

[G] Fulton and Harris: Representation Theory: A First Course – The first chapters make up an excellent introduction to the representation theory of finite groups; the later chapters to that of Lie groups and Lie algebras. Everything in this book is done with a mind to the student.

[G] Gelfand and Manin: Methods of Homological Algebra — A comprehensive, very modern homological algebra book, covering progress made in the subject since the 60’s.

[U] Hoffman and Kunze: Linear Algebra — One of the very few good books on linear algebra.

[G] Jacobson: Basic Algebra (Two Volumes) — A thorough and well-written text giving the basic results of almost every field of Abstract Algebra with many fascinating applications. Very dense and fast moving; this should not be your first book on Abstract Algebra. Out of print, but can be bought online.

[U] James and Liebeck: Representations and Characters of Groups – Very accessible introduction to the representation theory of groups. The focus is on finite groups, but there is a chapter on GL(2,q), two chapters on applications to group theory, and a chapter on the applications of character theory to molecular vibrations.

[U] Lang: Linear Algebra — Standard undergraduate book.

[U] Lang: Undergraduate Algebra – Concise, but easier than Herstein. Excellent for learning.

[G] Lang: Algebra – The standard reference for abstract algebra. Much more difficult to learn from than the above book, but indispensable as a reference.

[U] Rotman: An Introduction to Group Theory — A good, accessible reference.

[G] Serre: Complex Semisimple Lie Algebras – A strong and direct treatment of Lie algebra theory.

[U/G] Serre: Linear Representations of Finite Groups — A surprisingly introductory account of the basic theory of linear representations of finite groups, with many good examples and exercises.

[G] Shafarevich: Basic Notions of Algebra

[U] Stewart, Ian: Galois Theory — A good down to earth introduction to field extensions with lots of doable problems. Highly recommended.

[U] Uspensky: Theory of Equations — A book that brings together algebra, analysis, computation, etc.

[G] Zariski & Samuel: Commutative Algebra (2 Vols). Covers the algebra necessary to study algebraic geometry. Moves slowly and gives examples, but it can be hard to see the forest for the trees.

[U] Alon and Spencer: The Probabilistic Method — Goes into more depth than Spencer. Lots of neat examples and applications.

[U] Bondy and Murty: Graph Theory with Applications — Good explanations, easy to read.

[U/G] Bressoud: Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture — This is a history of the proof of the Alternating Sign Matrix Conjecture, including a full proof and many other elegant combinatorial results. This book does a great job of showing the interplay between observations, proofs and analogies in mathematical research. The writing style is very easy and casual, but the math is quite serious.

[U] Brylawski: Studies in Combinatorics — One of a super series edited by Rota and put out by the MAA.

[U] Chung and Graham: Erdos on Graphs — A compilation of Erdos’ work on graphs with a huge collection of open problems and their status. If you’re looking for a challenging and intriguing graph theory conjecture to try your hand at, or for the current status of most graph theory questions, look here.

[U] Cohen, D. I. A. : Introduction to Combinatorial Mathematics — One of the best introductions to combinatorics and combinatorial reasoning.

[U/G] Comtet: Advanced Combinatorics — Advanced and somewhat old-fashioned, but good when you have mastered the basics.

[U] Conway, John H.: On Numbers and Games — A more sophisticated and theoretical presentation of surreal numbers (and games) than Knuth’s book

[U] Graham, Knuth, and Patashnik: Concrete Mathematics: A Foundation for Computer Science — This book is an excellent account of discrete mathematics. A large number of high quality exercises with solutions.

[U] Graham, Spencer, Rothschild: Ramsey Theory — One of the clearest introductions to the subject.

[U] Ryser: Combinatorial Mathematics — Clearly written and very accessible.

[U] Spencer: Ten Lectures on the Probabilistic Method — Short, easy to read book on probabilistic combinatorics.

[U/G] Stanley: Enumerative Combinatorics, v. I and II — A graduate-level book, now the standard reference; it goes very deep, but it should be readable when you know the basics. It has lots of good problems, which are all given difficulty ratings, with solutions. EC2 contains 66 interpretations of the Catalan numbers.

[U] Trudeau: Introduction to Graph Theory — A very readable first book on graph theory.

[U] Wilf: Generatingfunctionology — An excellent introduction to generating functions.

[U] Wilson, Robin: Introduction to Graph Theory — Clearly written and very accessible. Nota Bene: Many of the journal articles in the subject are accessible. See for example the Journal of Combinatorial Theory.

[U] Armstrong: Basic Topology — An excellent introduction to the subject; treats both geometry and formalism.

[U] Arnold: Intuitive Concepts in Elementary Topology — Very intuitive; very accessible.

[G] Booß: Topology and Analysis: The Atiyah-Singer, etc. — Excellent introduction to the circle of ideas that makes up index theory, one of the hottest topics in modern geometry.

[G] Bredon: Topology and Geometry — Works out a lot of examples in cohomology theory. Grounded in geometry.

[U] Chinn, Steenrod: First concepts of topology. — A highly motivated introduction to the basic ideas of topology.

[G] Guillemin and Pollack: Differential Topology – Covers, among other things, many of the topics in Milnor, but with more depth; excellent exposition.

[G] Hatcher: Algebraic Topology — A beautiful, very geometric introduction to algebraic topology of all sorts, with lots of examples worked out. One of the first algebraic topology books to get it right, incorporating many modern techniques while keeping everything accessible.

[U/G] Hocking and Young: Topology — An excellent, compact reference for point-set topology, with quite a bit of algebraic topology thrown in as well.

[G] May: A Concise Course in Algebraic Topology – The title says it all. It contains everything there is to know about algebraic topology, including an extremely thorough treatment of homotopy theory. It is also frighteningly concise and modern; as a result it prepares you to start reading modern research papers directly. Not for the faint of heart, but a very rewarding read if you give it the patience it deserves.

[U/G] Massey: A Basic Course in Algebraic Topology — Readable, lots of examples. Combines much of his Algebraic Topology: An Introduction and Singular Homology Theory.

[U] Milnor: Topology from the Differentiable Viewpoint — A good introduction to differential topology. Very concise and elegant.

[U] Munkres: Topology: A First Course — The standard introduction to point-set topology and a bit of algebraic topology; very well written. The new, expanded second edition is called Topology.

[G] Spanier: Algebraic Topology — Covers lots of good topics in algebraic topology; rather abstract and categorical.

[U] Steen, Lynn Arthur: Counterexamples in Topology – An excellent source from which to see exactly why all those assumptions are necessary for some theorems. One of the best books out there for building intuition.

[G] Vick: Homology Theory — A good place to learn homology theory.

[G] Adams: Lectures on Lie Groups – A clear and concise introduction to the theory of compact Lie groups. Bypasses Lie algebra theory.

[G] Almgren: Plateau’s Problem: An Invitation to Varifold Geometry — A short book on a single topic. Read after knowing some of the basics.

[U] Coxeter & Greitzer: Geometry Revisited — Coxeter and Greitzer give an exciting treatment of Euclidean geometry picking up where high school left off. The exercises are interesting and engaging like the text.

[U/G] Coxeter: Introduction to Geometry — This is geometry as it ought to be done.

[G] Dubrovin, Fomenko, Novikov: Modern Geometry — Methods and Applications. — 3 volumes. A comprehensive treatment of topology and geometry that never loses sight of visual intuition. Requires college-level mathematical maturity.

[U] Greenberg, Marvin: Euclidean and Non-Euclidean Geometry: Development and History — An excellent introduction with a lot of motivation and history.

[G] Hartshorne: Algebraic Geometry — A standard reference for the scheme-theoretic viewpoint of algebraic geometry. Contains many exercises, as notorious as they are pedagogically sound. Should be read in conjunction with a more classically-minded text, such as Shafarevich’s Basic Algebraic Geometry, Mumford’s Complex Projective Varieties, or Harris’s Algebraic Geometry: A First Course.

[G] Griffiths and Harris: Principles of Algebraic Geometry — Algebraic geometry from an entirely analytic point of view. Very classical, very beautiful, and very difficult; fundamentally different in flavor, texture, and color from the other books on this list.

[U] Hilbert and Cohn-Vossen: Geometry and the Imagination — Hard to find, but great for getting geometric insight.

[U] Kedlaya: Packet on Euclidean Geometry — Has great problems and tools for Euclidean Geometry.

[U] Madsen and Tornheave: From Calculus to Cohomology – Has a very modern and sure-footed style, yet remains extremely accessible. Highly recommended as a first book on manifolds.

[G] Mumford: The Red Book of Varieties and Schemes — The only book that tells you why schemes are the correct objects to study in algebraic geometry. Lots of examples and explanation of very advanced topics from one of the best expositors ever.

[G] Thurston: Three-Dimensional Topology and Geometry. — Visual intuition is stressed over rigor. Lots of great exercises.

[U] Yaglom: Geometric Transformations — A delightful book.

[U/G] Ahlfors: Complex Analysis — The standard first graduate textbook.

[U] Apostol: Calculus (Vols. 1 and 2) — The second-best calculus text out there. Also contains several-variable calculus and some linear algebra.

[U] Apostol: Mathematical Analysis – More difficult and more abstract than Buck, but valuable and readable. It continues where Buck leaves off, but should be accessible after any good elementary course.

[U] Boas: A Primer of Real Functions — A fun book on real analysis; focuses on giving interesting examples and cool facts rather than difficult machinery.

[U] Buck: Advanced Calculus — A classic. Highly recommended.

[U/G] Cartan: Elementary Theory of Analytic Functions of One or Several Complex Variables — Cartan's book is a brisk and rigorous introduction to complex analysis. It is complex analysis absolutely done right. Perfect for those comfortable with a fairly high level of abstraction.

[U] Conway, John B.: Functions of One Complex Variable — Another excellent text on complex analysis, but more difficult than Ahlfors.

[U] deBruijn: Asymptotic Methods in Analysis — Explains how to compute the rates of growth of solutions to functions defined by integrals, differential equations or implicitly with many examples. Although it does not cover applications to combinatorics, combinatorialists will find it helpful.

[U] Gelbaum, Bernard R.: Counterexamples in Analysis -- An excellent source from which to see exactly why all those assumptions are necessary for some theorems. One of the best books out there for building intuition.

[U] Keisler, H. Jerome: Elementary Calculus: An Infinitesimal Approach – Keisler's book is an alternative approach to Calculus using the idea of an infinitesimal rather than epsilon-delta. It does not assume any calculus and is available for free online.

[U] Knopp: Sequences and Series — A beautiful little book.

[U] Knopp: Theory of Functions (Three Volumes and Two Problem Books) — An introduction to functions of a complex variable; requires a high degree of sophistication; the problem books are especially good.

[U] Kuga: Galois’ Dream — A really fun book introducing groups, differential equations, and one of their connections, the surprising world of differential Galois theory.

[U] Pugh, Charles Chapman: Real Mathematical Analysis -- An absolutely fantastic introduction to Analysis, it has excellent exposition and is full of great examples and over 500 (good) exercises. A counterpart to Rudin, Pugh always builds up machinery first and uses it to provide very clear proofs that grant a good sense of "why" something is true. Judging by page count and the amount of material it covers, it seems that it must be as concise as Rudin, yet it reads very easily.

[U] Rudin: Principles of Mathematical Analysis — An accessible, traditional introduction to hard analysis. The first half is good, but the last few chapters are not a good place to learn about differential forms and Lebesgue integration.

[G] Rudin: Real and Complex Analysis — An excellent text combining real and complex analysis, starting with a good introduction to measure and integration theory.

[U] Schey: Div, Grad, Curl and all that — An excellent book for understanding multivariable calculus.

[U/G] Shakarchi and Stein: Princeton Lectures in Analysis —The new standard textbook for Fourier analysis (volume I), complex analysis (volume II), and real analysis and measure theory (volume III). When volume IV comes out, it will have a hodgepodge of topics in modern analysis, such as distribution theory. Blows all other analysis books out of the water in quality of exposition, clarity, and depth.

[U] Spivak: Calculus — Spivak takes the honors as the best introductory calculus text: a thorough treatment of the subject beautifully written.

[U] Spivak: Calculus on Manifolds — This standard reference is a classical approach to multivariable calculus, manifolds and differential forms, making them painful, ugly and confusing.

[G] Stein: Introduction to Fourier Analysis on Euclidean Spaces – A rigorous introduction to Fourier analysis on Euclidean spaces, with an emphasis on operators and interpolation.

[U/G] Whittaker & Watson: Modern Analysis — The classic text on complex variables and transcendental functions, includes an excellent section on the Jacobi theta functions. The beginning is a very thorough, though nonstandard, course on complex analysis.

[U] Adams and Guillemin: Measure Theory and Probability – A good introduction to measure theory, emphasizing applications to probability theory. The second half deals with Fourier analysis and Lebesgue integration.

[G] Dudley: Real Analyis and Probability — A fairly sophisticated and advanced introduction. Very complete and gives concise proofs.

[U] Feller: An Introduction to Probability Theory and its Applications, Fifth Edition. — The standard reference.

[U] Larsen and Marx: An Introduction to Mathematical Statistics and its Applications — Fun to read.

[U] Mosteller: Fifty Challenging Problems in Probability — Great problems, highly recommended.

[U] Halmos: Naive Set Theory — A classic written by a great expositor.

[U] Delong: A Profile of Mathematical Logic — Very accessible.

[U] Ebbinghaus, Flum, and Thomas: Mathematical Logic — A good introduction to first-order logic, Gödel’s Incompleteness Theorem, and Model Theory.

[U] Enderton, Herbert: A Mathematical Introduction to Logic -- Enderton's strong points are that the book is concise, very simple, and does a great job of connecting logic with other fields of math (such as Algebra and Analysis).

[U] Hofstadter, D.: Gödel, Escher, Bach, an Eternal Golden Braid -- A witty guide to Gödel’s proof and related philosophical ideas.

[G] Krivine: Introduction to Axiomatic Set Theory -- A complete introduction; very dense.

[G] Jech: Set Theory — Includes the proof of Cohen’s independence results.

[U/G] Mendelson: Introduction to Mathematical Logic — Best all-around introductory text.

[U] Nagel and Newman, J.: Gödel’s Proof — Not hard to read, and no background required.

[U] Roitman, Judith: Introduction to Modern Set Theory — An introduction to axiomatic set theory, ordinals, cardinals. Elementary and very readable.

[U] Russell: Introduction to Mathematical Philosophy — A classic and accessible, though somewhat dated introduction.

[U/G] Smorynski: Logical Number Theory I — Does basic logic with a heavy focus on Number Theory applications; includes clear proofs of Gödel’s theorem and Hilbert’s Tenth Problem (that there exists no algorithm to decide whether a Diophantine equation is solvable.) Volume II was never published.

[U] Smullyan: Forever Undecided : A Puzzle Guide to Gödel — This book uses liar/truthteller puzzles to walk you through Gödel’s proof.

[U] Vaught: Set Theory — A readable, thorough and rigorous undergraduate introduction.

Abelson and Sussman: Structure and Interpretation of Computer Programs — A sophisticated view of programming languages. Teaches in Scheme, a dialect of Lisp. You should not try to learn Scheme from this book. This books describes how to compute (with a computer) numerical values of mathematical expressions. Read this either to find out how Mathematica and cousins work, or when you need something they won’t do. Although a new edition came out in the 90’s, very little has been changed since the 1970 version.

Corman, Leiserson, Rivest, and Stein: Algorithms 2nd ed. — A good, comprehensive account of algorithms.

Hillis, W. Daniel: The Connection Machine — The ideas leading up to the CM, before a concrete one existed. An introduction to some ideas in parallel processing.

Horowitz and Sahni: Fundamentals of Computer Algorithms.

Hwang, Kai, and Briggs, Faye: Computer Architecture and Parallel Processing — Very technical, but full of creative ideas linking mathematics, computers and engineering. A “classic” in computer science engineering.

Knuth: The Art of Computer Programming 3rd ed. — Excellent, especially the chapter on arithmetic. The problems are ranked by difficulty. (There are only three volumes, although some editions claim that others exist.)

Menezes: Handbook of Applied Cryptography – This is a highly mathematical guide to implementing cryptographic protocols, focusing on public-key cryptography and key management. Available at http://www.cacr.math.uwaterloo.ca/hac/

Papadimitriou: Computational Complexity — An introduction to the study of the complexity of algorithms, including NP-completeness and related ideas.

Schneier: Applied Cryptography 2nd ed. – A classic in the field of cryptography. Gives detailed descriptions of major cryptographic and security protocols.

Sedgewick: Algorithms — Excellent source of code for many different algorithms and data structures. Available in editions for Pascal, C, C++, Java, and Modula-3.

Sipser: Introduction to the Theory of Computation – Very readable introduction to the theory of computation, including regular languages, context-free grammars, Turing machines, and complexity.

Stinson: Cryptography: Theory and Practice – An thorough survey of modern cryptography and cryptanalysis, covering information theoretic entropy and perfect security, DES, secure key exchange, public key crypto (RSA and other systems), signatures, and zero-knowledge proofs.

Walter: The Secret Guide to Computers — Call (617) 666-2666 to order from the author. The guide to everything from software to hardware. Well-written full with history and jokes.

Albers and Alexanderson, ed.: Mathematical People: Profiles and Interviews

Albers, Alexanderson, and Reid, ed.: More Mathematical People: Contemporary Conversations

Baum: The Calculating Passion of Ada Byron

Bell: Men of Mathematics — Readable and very interesting, though opinionated and often patently false.

Halmos: I Want To Be a Mathematician: An Automathography

Hardy: A Mathematician’s Apology — Thought provoking.

Hoffman: The Man Who Loved Only Numbers — An Erdös biography.

Infeld: Whom the gods love — The story of Evariste Galois

Kanigel: The Man Who Knew Infinity — A biography of Ramanujan, also has a good deal on Hardy.

Klein, Morris: Mathematical Thought from Ancient to Modern Times

Koestler: The Watershed — A biography of Johannes Kepler

Kovalevskaia: A Russian Childhood

Osen: Women in Mathematics

Reid: Hilbert-Courant

Reid: Courant in Gottingen and New York

Schechter: My Brain is Open — Another Erdös biography.

Abbot: Flatland — An imaginative excursion into the third dimension from the plane.

Barnsley: Fractals Everywhere — Also presents the mathematics behind the pretty pictures.

Benaceraff and Putnam: Philosophy of Mathematics: Selected Readings — Only if you’re serious about thinking deeply.

Berlekamp, Conway, John H., and Guy: Winning Ways — A novel view of the mathematical theory of combinatorial games. It’s the standard reference work, but it’s also an easy to read introduction with lots of pretty pictures and neat games.

Burger, D.: Sphereland — Nice sequel to Flatland, though somewhat of a letdown.

Burger, E. & Starbind: The Heart of Mathematics — A light-hearted, yet mathematically precise, survey of topics throughout mathematics. Clearly written, easy to follow and in fact a joy to read.

Courant and Robbins: What is Mathematics? — Interesting and stimulating.

Devaney: An Introduction to Chaotic Dynamical Systems — The mathematics behind the pretty pictures, at a level assuming calculus and linear algebra.

Dorrie: 100 Great Problems of Elementary Mathematics — You’ll be surprised what can be done with high school math!

Dunham, William: Journey Through Genius: The Great Theorems of Mathematics — An easy introduction to the history of math (fun theorems!)

Hadamard: The Psychology of Invention in the Mathematical Field — An investigation into the roots of mathematical creativity. Also published as The Mathematician’s Mind.

Honsberger: Mathematical Gems I and II

Mathematical Plums

Mathematical Morsels — Filled with lovely elementary tidbits.

Knuth: Surreal Numbers — Perhaps the only mathematical discovery first published as a work of fiction.

Lakatos: Proofs and Refutations: The Logic of Mathematical Discovery — A classic book on mathematical heurisitics and methodology.

Lawvere & Schanuel: Conceptual Mathematics: A First Introduction to Categories — A highly intuitive introduction to category theory.

Mazur: Imagining Numbers (Particularly the Square Root of Minus Fifteen) – A cute book on number theory for the layman by Professor Stevens’s Doktorvater.

Polya: How to Solve It — A wonderful little book on the problem of how to solve problems. Along similar lines are Polya’s Mathematics and Plausible Reasoning and Mathematical Discovery.

Sloane: The Encyclopedia of Integer Sequences: A listing of more than 200,000 integer sequences that arise in mathematical problems and the problems in which they show up.